Integration of model structures

Let $M$ be a model category and $F∶ M \to \underline{\mathrm{ModCat}}$ a relative functor. If $F$ satisﬁes a "strengthened relativeness" condition, we deﬁne a model structure on the Grothendieck construction $\int_M F$ called the integral model structure. This model structure is “homotopy-invariant” in that for two such functors $F, F': M \to \underline{\mathrm{ModCat}}$, a pseudo-natural transformation $\Phi ∶ F \Rightarrow F'$ which is an object-wise Quillen equivalence induces a Quillen equivalence between the corresponding integral structures:$$\int_M F \simeq \int_M F'$$

As applications, we describe the integral model structure that arises from several functors. These include the functor $B∶ sGp \to \underline{\mathrm{ModCat}}$ given by the projective model structure on $G$-spaces, $G \mapsto S^G$ and the functor $(\bullet)-\mathrm{LMod}: \mathrm{Alg}(C) \to \underline{\mathrm{ModCat}}$ given by the model structure of (left) modules over an algebra object in a suitable monoidal model category $C$, satisfying the assumptions of Schwede and Shipley. This is joint work with Y. Harpaz.